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I'm trying to launch a few programs (primarily audacity) from buttons on a flash page.
Not designed to do that.
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From what I've been able to find online, AIR and Flex can launch executables in windows, but not in nix.
I wouldn't depend on that even in Windows. Cow-sized security holes aside, it'd mean relying on a feature likely to disappear the first time anyone abuses it, or at least buried under n+1 layers of "are you really sure you're really sure" the way ActiveX ended up.
There's not anything running locally for your flash script to connect to either, unless you run it yourself. You could set up a webserver that allows connections
only on localhost, install sudo, and allow your webserver to run
sudo -i -u username /path/to/command so it can pop up windows on you when you load a PHP or CGI script. This is still a gaping security hole in that it allows anything running on your system to run arbitrary programs as you, but at least you're protected from the network at large.
In the end, though? Don't: This is a bad idea anywhere. Pick a language suitable for the job and use that.